MinT - Best Known (118−57, 118, s)-Nets in Base 5

Best Known (118−57, 118, s)-Nets in Base 5

(118−57, 118, 84)-Net over F₅ — Constructive and digital

Digital (61, 118, 84)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 30, 12)-net over F₅, using
  - net from sequence [i] based on digital (2, 11)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 2 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 88, 72)-net over F₅, using
  - net from sequence [i] based on digital (31, 71)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
      - an explicitly constructive algebraic function field [i]

(118−57, 118, 134)-Net over F₅ — Digital

Digital (61, 118, 134)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(118−57, 118, 2332)-Net in Base 5 — Upper bound on s

There is no (61, 118, 2333)-net in base 5, because

1 times m-reduction [i] would yield (61, 117, 2333)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 6042 366444 971547 267921 613972 464532 182058 187583 067657 731997 553846 197801 763052 746385 > 5¹¹⁷ [i]